where \(Z_{\rm depro}\)  is the deprojected abundance profile, \(A_{\rm Fe}\)  is the atomic weight of iron, and \(m_{\rm H}\)  is the atomic unit mass.
The hydrogen density \(n_{\rm H}\)  is derived from the gas density \(n_{\rm gas}\)  through the usual relation \( n_{\rm gas}= \left(1 + n_e/n_{\rm H} \right)n_{\rm H} = 2.21 n_{\rm H}\) , where \(n_e\)  is the electron density; \(n_{\rm gas}\)  is obtained through deprojection. In the latter case \(M_{\rm Fe,500}=M_{\rm Fe} (<r_{500})\) .
For the measurement of \(M^{\rm star}_{\rm Fe,500}\) 
it is usually assumed that the average iron abundance in clusters and groups stars is solar [1]}. This iron abundance of the stars is then multiplied by the total stellar mass enclosed within \(r_{500}\) . The latter value can again be estimated through two approaches. The first one performs a flux measurement for each galaxy in a given optical band and calculates the mass of the galaxy through the Spectral Energy Distribution (SED) fitting, the mass of the galaxies are then summed together [2]}, [3]}. The second approach calculates the total luminosity in a given optical band by integrating the luminosity function of the red cluster galaxies, summing the contribution of the BCG and possibly of ICL and then multiply for an assumed stellar-mass-to-light ratio in the same optical band [4]}.
Both quantities are deprojected assuming a generalized or a simple Navarro-Frenk-White (NFW) distribution for the galaxy and optical light distribution.
These observational estimates can be compared with the expected theoretical estimate based on the current understanding of stellar nucleosynthesis. We take the values reported in [3]} for the \(\mathcal {Y}_{\rm Fe}\)  based on the derivation by [4]} and [7]}. \(\mathcal {Y}_{\rm Fe}\)  is computed as the product of the Fe mass produced by a SN explosion, \(y\) , and the number of SN events produced per unit mass of gas turned into stars, \(k\) .
Both contributions from Ia and CC SN are considered. Thus, \(\mathcal {Y}_{\rm Fe}\)  can be written as:
\(\mathcal {Y}_{\rm Fe} = y_{\rm Ia} \cdot k_{\rm Ia} + y_{\rm CC} \cdot k_{\rm CC},\) 
